Your explanation is not quite clear . Moreover , if 4x + 9/x decreases when 0 < x <= 1 , how does it imply that 4sin^2(x) + 9/sin^2(x) has a minimum when sin^2(x) = 1 ? You'll have to state the reason if any . And further more , you've found out the min value of 4sin^2(x) + 9/sin^2(x) which is not exactly what i posted . Please read the question carefully . Graph of sin and cosec are distinctly different!!

if 4x + 9/x decreases when 0 < x <= 1 , how does it imply that 4sin^2(x) + 9/sin^2(x) has a minimum when sin^2(x) = 1 ?

The function sin^2(x) assumes all values between 0 and 1 and only them. Therefore, the minimum of 4sin^2(x) + 9/sin^2(x) on all real line equals the minimum of 4x + 9/x when 0 < x <= 1. Moreover, the minimum of 4x + 9/x on this interval is assumed when x = 1, so the minimum of 4sin^2(x) + 9/sin^2(x) is assumed when sin^2(x) = 1. If you need more explanation, please say if this is not intuitively clear or if you are looking for formal reasons, such as how this can be deduced from the axioms of real numbers. (Deduction from axioms, even though it is the most air-tight proof, is often not the clearest.)

And further more , you've found out the min value of 4sin^2(x) + 9/sin^2(x) which is not exactly what i posted .

emakarov did state the reason! He said that 4x+ 9/x is decreasing for 0< x< 1 and suggested how you could see that for yourself. Since 4(1)+ 9/(1= 13, the minimum value of 4x+ 9/x, or any 4f(x)+ 9/f(x) where f(x) lies between 0 and 1.

The function sin^2(x) assumes all values between 0 and 1 and only them. Therefore, the minimum of 4sin^2(x) + 9/sin^2(x) on all real line equals the minimum of 4x + 9/x when 0 < x <= 1. Moreover, the minimum of 4x + 9/x on this interval is assumed when x = 1, so the minimum of 4sin^2(x) + 9/sin^2(x) is assumed when sin^2(x) = 1. If you need more explanation, please say if this is not intuitively clear or if you are looking for formal reasons, such as how this can be deduced from the axioms of real numbers. (Deduction from axioms, even though it is the most air-tight proof, is often not the clearest.)

What is the difference?

Beautiful explanation ! Totally clear now .

But are you aware ( please let me know! ) why such a , what should I say , Paradox is happening when we use the perfect square method ?